***Question*** : are the continuous characters of the form 

 - $\eta : \mathbb{Z}_p^* \to \mathbb{Z}_p^*$, or
 - $\eta : (1+p\mathbb{Z}_p)^{\times} \to \mathbb{Z}_p^*$ (i.e., on the principal units in $\mathbb{Z}_p^*$)

well understood? Can such characters be classified in either case ?